Sequent and Hypersequent Calculi for Abelian and Lukasiewicz Logics

We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for L. These include: hypersequent calculi for A and L and terminating versions of these calculi; labelled single sequent calculi for A and L of complexity co-NP; unlabelled single sequent calculi for A and L.Comment: 35 pages, 1 figur

On the difficulty of presenting finitely presentable groups

We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of the finitely presentable subgroups.Comment: Final version. To appear in GGD volume dedicated to Fritz Grunewal

Rolf Mantel and the Computability of General Equilibria: On the Origins of the Sonnenschein-Mantel-Debreu Theorem

In this brief paper we revise the original motivations of Rolf Mantel to pursue a proof of Sonnenschein´s conjecture. We contend that his work on computational models of general equilibrium lead him to seek an alternative to the usual fixed point theorems used in proofs of existence. Confronted with a paper of Uzawa and his own experience in programming a national planning system he found that the use of theorems like Brouwer´s and Kakutani´s was unavoidable. To check out whether Uzawa was right he sought to find out whether the only properties required of excess demand functions to ensure the existence of equilibria in competitive markets were continuity, homogeneity and Walras´ law. In 1974, he found that this was actually the case. We will see that this result and his interpretation were informed by Mantel´s interest in economic development and planning.Fil: Tohmé, Fernando

2-Nested Simulation is not Finitely Equationally Axiomatizable

2-nested simulation was introduced by Groote and Vaandrager [10] as the coarsest equivalence included in completed trace equivalence for which the tyft/tyxt format is a congruence format. In the lineartime-branching time spectrum of van Glabbeek [8], 2-nested simulationis one of the few equivalences for which no finite equational axiomatization is presented. In this paper we prove that such an axiomatizationdoes not exist for 2-nested simulation.Keywords: Concurrency, process algebra, basic CCS, 2-nested simulation, equational logic, complete axiomatizations

Computability for the absolute Galois group of Q\mathbb{Q}

The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation and use it to address several effectiveness questions about Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$: the difficulty of computing Skolem functions for this group, the arithmetical complexity of various definable subsets of the group, and the extent to which countable subgroups defined by complexity (such as the group of all computable automorphisms of the algebraic closure $\overline{\mathbb{Q}}$) may be elementary subgroups of the overall group

Grothendieck ring of semialgebraic formulas and motivic real Milnor fibres

We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points satisfying this formula and contains as a ring the usual Grothendieck ring of real algebraic formulas. We give a realization of our ring that allows to express a class as a Z[1/2]- linear combination of classes of real algebraic formulas, so this realization gives rise to a notion of virtual Poincar\'e polynomial for basic semialgebraic formulas. We then define zeta functions with coefficients in our ring, built on semialgebraic formulas in arc spaces. We show that they are rational and relate them to the topology of real Milnor fibres.Comment: 30 pages, 1 figur

Nested Semantics over Finite Trees are Equationally Hard

This paper studies nested simulation and nested trace semantics over the language BCCSP, a basic formalism to express finite process behaviour. It is shown that none of these semantics affords finite (in)equational axiomatizations over BCCSP. In particular, for each of the nested semantics studied in this paper, the collection of sound, closed (in)equations over a singleton action set is not finitely based

Interprocedural program analysis using visibly pushdown Kleene algebra

Les analyses interprocédurales automatiques de programmes qui sont basées sur des théories mathématiques rigoureuses sont complexes à réaliser, mais elles sont d'excellents outils pour augmenter notre conance envers les comportements possibles d'un programme. Les méthodes classiques pour réaliser ces analyses sont l'analyse de modè- les, l'interprétation abstraite et la démonstration automatique de théorèmes. La base d'un démonstrateur automatique de théorèmes est une logique ou une algèbre et le choix de celle-ci a un impact sur la complexité de trouver une preuve pour un théorème donné. Cette dissertation développe un formalisme algébrique concis pouvant être utilisé en démonstration automatique de théorèmes. Ce formalisme est appellé algèbre de Kleene à pile visible. Cette dissertation explique comment ce formalisme peut être utilisé pour réaliser des analyses interprocédurales de programmes, comme des vérications formelles et des vérications d'optimisations efectuées par des compilateurs. Cette dissertation apporte aussi des preuves que ces analyses pourraient être automatisées. L'algèbre de Kleene à pile visible est une extension de l'algèbre de Kleene, un excellent formalisme pour réaliser des analyses intraprocédurales de programmes. En bref, l'algèbre de Kleene est la théorie algébrique des automates nis et des expressions régulières. Donc, cette algèbre à elle seule n'est pas appropriée pour faire des analyses interprocédurales de programmes car la puissance des langages non contextuels est souvent nécessaire pour représenter le lot de contrôle d'un tel programme. L'algèbre de Kleene à pile visible étend celle-ci en lui ajoutant une famille d'opérateurs de plus petit point xe qui est basée sur une restriction des grammaires non contextuelles. En fait, cette algèbre axiomatise exactement la théorie équationnelle des langages à pile visibles. Ces langages sont une sous-classe des langages non contextuels et ont été dénis par Alur et Madhusudan pour faire de l'analyse de modèles. La complexité résultante de la théorie équationnelle de l'algèbre proposée est EXPTIME-complète.Automatic interprocedural program analyses based on rigorous mathematical theories are complex to do, but they are great tools to increase our condence in the behaviour of a program. Classical ways of doing them is either by model checking, by abstract interpretation or by automated theorem proving. The basis of an automated theorem prover is a logic or an algebra and the choice of this basis will have an impact in the complexity of nding a proof for a given theorem. This dissertation develops a lightweight algebraic formalism for the automated theorem proving approach. This formalism is called visibly pushdown Kleene algebra. This dissertation explains how to do some interprocedural program analyses, like formal veri cation and verication of compiler optimizations, with this formalism. Evidence is provided that the analyses can be automated. The proposed algebraic formalism is an extension of Kleene algebra, a formalism for doing intraprocedural program analyses. In a nutshell, Kleene algebra is the algebraic theory of nite automata and regular expressions. So, Kleene algebra alone is not well suited to do interprocedural program analyses, where the power of context-free languages is often needed to represent the control flow of a program. Visibly pushdown Kleene algebra extends Kleene algebra by adding a family of implicit least xed point operators based on a restriction of context-free grammars. In fact, visibly pushdown Kleene algebra axiomatises exactly the equational theory of visibly pushdown languages. Visibly pushdown languages are a subclass of context-free languages dened by Alur and Madhusudan in the model checking framework to model check interprocedural programs while remaining decidable. The resulting complexity of the equational theory of visibly pushdown Kleene algebra is EXPTIME-complete whereas that of Kleene algebra is PSPACE-complete